Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{p^2 - 5p}{3p + 24} \times \dfrac{5p - 5}{p^2 - 6p + 5} $
Solution: First factor the quadratic. $r = \dfrac{p^2 - 5p}{3p + 24} \times \dfrac{5p - 5}{(p - 5)(p - 1)} $ Then factor out any other terms. $r = \dfrac{p(p - 5)}{3(p + 8)} \times \dfrac{5(p - 1)}{(p - 5)(p - 1)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac{ p(p - 5) \times 5(p - 1) } { 3(p + 8) \times (p - 5)(p - 1) } $ $r = \dfrac{ 5p(p - 5)(p - 1)}{ 3(p + 8)(p - 5)(p - 1)} $ Notice that $(p - 1)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac{ 5p\cancel{(p - 5)}(p - 1)}{ 3(p + 8)\cancel{(p - 5)}(p - 1)} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $r = \dfrac{ 5p\cancel{(p - 5)}\cancel{(p - 1)}}{ 3(p + 8)\cancel{(p - 5)}\cancel{(p - 1)}} $ We are dividing by $p - 1$ , so $p - 1 \neq 0$ Therefore, $p \neq 1$ $r = \dfrac{5p}{3(p + 8)} ; \space p \neq 5 ; \space p \neq 1 $